Title: Start-Up of FEL Oscillator from Shot Noise

Abstract

In free-electron laser (FEL) oscillators, as inself-amplified spontaneous emission (SASE) FELs, the buildup of cavitypower starts from shot noise resulting from the discreteness ofelectronic charge. It is important to do the start-up analysis for thebuild-up of cavity power in order to fix the macropulse width from theelectron accelerator such that the system reaches saturation. In thispaper, we use the time-dependent simulation code GINGER [1]toperformthis analysis. We present results of this analysis for theparameters of the Compact Ultrafast TErahertz FEL (CUTE-FEL) [2]beingbuilt atRRCAT.

We describe the radiation properties of an x-ray free-electron laser (FEL) oscillator, beginning with its start-up from noise through saturation. We first decompose the initially chaotic undulator radiation into the growing longitudinal modes of the composite system consisting of the electron beam, the undulator, and the Bragg mirror resonator cavity. Because the radiation initially comprises several modes whose growth rates are comparable, we find that only after many oscillator passes is the output pulse dominantly characterized by the lowest-order Gaussian mode. We verify our analytic results with a novel, reduced one-dimensional FEL code (derived in the text), and with two-dimensionalmore » FEL simulations. Understanding the full longitudinal structure during the initial amplification will be critical in assessing the tolerances on the electron beam, undulator, and optical cavity required for robust operation.« less

We describe the radiation properties of an x-ray free-electron laser (FEL) oscillator, beginning with its start-up from noise through saturation. We first decompose the initially chaotic undulator radiation into the growing longitudinal modes of the composite system consisting of the electron beam, the undulator, and the Bragg mirror resonator cavity. Because the radiation initially comprises several modes whose growth rates are comparable, we find that only after many oscillator passes is the output pulse dominantly characterized by the lowest-order Gaussian mode. We verify our analytic results with a novel, reduced one-dimensional FEL code (derived in the text), and with two-dimensionalmore » FEL simulations. Understanding the full longitudinal structure during the initial amplification will be critical in assessing the tolerances on the electron beam, undulator, and optical cavity required for robust operation.« less

Linearized Vlasov-Maxwell equations are used to derive a partial differential equation determining the 3-dimensional slowly varying envelope function of the radiated electric field. The equation is solve analytically. From the correlation function of the electric field averaged over the stochastic ensemble describing the initial shot noise in the beam, we compute the longitudinal and transverse correlation lengths sigma/sub parallel/ and sigma/sub perpendicular/. The radiated power S per unti cross-sectional area of the electron beam is S = rhoS/sub e//9n/sub 0/V/sub c/ exp(..sqrt..3 4..pi..N/sub w/rho), where V/sub c/ = (2..pi..)/sup 3/2/ sigma/sub parallel/ sigma/sub perpendicular//sup 2/ is the coherence volume, n/submore » 0/ the electron density, S/sub e/ = (..gamma../sub 0/mc/sup 2/) n/sub 0/c the power per unit area in the electron beam, N/sub w/ the number of wiggler periods and rho the Pierce parameter. The angular distribution of the radiation is characterized by the Gaussian factor exp(-theta/sup 2//2sigma/sub theta/ /sup 2/), where 2..pi..sigma/sub theta/sigma/sub perpendicular/ = lambda (radiated wavelength). Our analysis is applicable for wiggler length L = N/sub w/lambda/sub w/ long enough for the exponential regime to be reached, but short enough so that L sigma/sub theta/approx.« less

Linearized Vlasov-Maxwell equations are used to derive a partial differential equation determining the 3-dimensional slowly varying envelope function of the radiated electric field. The equation is solved analytically. From the correlation function of the electric field averaged over the stochastic ansemble describing the initial shot noise in the beam, we compute the longitudinal and transverse correlation lengths sigma/sub L/ and sigma/sub T/. The radiated power S per unit cross-sectional area of the the electron beam is determined. Our analysis is applicable for wiggler length L = N/sub w/lambda/sub w/ long enough for the exponential regime to be researched, but shortmore » enough so that L sigma/sub theta/ less than or equal to a, the electron beam radius. 6 refs.« less

A one-dimensional linear analysis is presented of the start-up of an FEL oscillator. The model treats electron beam pulses of arbitrary shape and many significant three dimensional effects are included heuristically. A closed equation is derived for the ensemble averaged electromagnetic energy density matrix which contains the spontaneous emission as a source, and which represents the small gain per pass and losses via coefficient matrices. Numerical solutions are compared with the Stanford experimental results.